We know that \(\Phi _k\)  is a concave function of \(\mathbf {e}\) . Hence, \(\varphi _k W \log _2(1 + \frac{e_k}{\varphi _k} \gamma _k) \)  is the perspective of \(\Phi _k\)[1]}. As the perspective operation preserves convexity, it is a concave function of \(e_k\)  and \(\varphi _k\) . Then, we can observe that \(R_{k}(\tau _k,\varphi _k,e_k)\)  is a concave function of \(\tau _k, \varphi _k\)  and \(e_k\)  since the summation of a linear function and a concave function is a concave function. As \(R(\mathbf {\tau }, \mathbf {\varphi }, \mathbf {e}) = \min \limits _{k}{R_{k}(\tau _k,\varphi _k,e_k)}\)  is the pointwise minimum of K concave functions, \(R(\mathbf {\tau }, \mathbf {\varphi }, \mathbf {e})\)  is a jointly concave function of \(\mathbf {\tau }\) , \(\mathbf {\varphi }\) , and \(\mathbf {e}\) .
